Math Insight

When Green's theorem applies

Suggested background

Green's theorem
provides another way to calculate
\begin{align*}
\dlint
\end{align*}
that you can use instead of calculating the line integral directly.
However, some common mistakes involve using Green's theorem to
attempt to calculate line integrals where it doesn't even apply.

First, Green's theorem works only for the case where $\dlc$
is a simpleclosed curve.
If $\dlc$ is an
open curve, please
don't even think about using Green's theorem.

If you think of the idea of Green's theorem in terms of circulation,
you won't make this mistake. Green's
theorem converts the line integral to a double integral of the
microscopic circulation. The double integral is taken over the region
$\dlr$ inside the path. Only closed paths have a region $\dlr$ inside them.
The idea of circulation makes sense only for closed paths. (And the line integral represents circulation around the region only if is simple.)

So if you are asked to compute the integral
\begin{align*}
\int_\dlc y\, dx + xy \, dy
\end{align*}
where $\dlc$ is the line from $(0,1)$ to $(1,1)$, can you use Green's
theorem? No, because $\dlc$ is an open curve.

Second, Green's theorem can be used only for vector fields in
two dimensions, such as the $\dlvf(x,y)= (y, xy)$ of the
above example. It cannot be used for vector fields in three
dimensions. So, don't bother with Green's theorem if you are given an
integral like
\begin{align*}
\int_\dlc z\, dx + xy \, dy - yz \, dz
\end{align*}
even if $\dlc$ is a closed path.

Why a whole page just to stress these simple ideas?
Because students frequently
under pressure try to use Green's theorem when it doesn't apply.
Maybe now that you've read this warning, you'll resist that temptation.